A CLASS OF FINITE ELEMENTS FOR NONLINEAR ANALYSIS OF COMPOSITE BEAMS Quang Huy NGUYEN INSA de Rennes – Structural Engineering Research Group 20 Avenue des Buttes de Coësmes, 35043 Rennes Cedex, France [email protected] Mohammed HJIAJ INSA de Rennes – Structural Engineering Research Group 20 Avenue des Buttes de Coësmes, 35043 Rennes Cedex, France [email protected] Brian UY School of Engineering, University of Western Sydney Penrith South DC NSW 1797, Australia [email protected] Samy GUEZOULI INSA de Rennes – Structural Engineering Research Group 20 Avenue des Buttes de Coësmes, 35043 Rennes Cedex, France [email protected]

ABSTRACT In this paper, we investigate the performance of three finite element formulations for continuous composite beam analysis. These formulations include displacement-based, force-based and mixed models. Specific attention is devoted on how to take into account discrete connection which seems, in some cases, to be more representative of the actual behavior. The nonlinear behavior of concrete is modeled using a softening plasticity model. The parameters of the models are obtained by matching the uniaxial stress-strain relationship provided by the CEB-FIP 90 model. Tension-stiffening is taken into account as well. A consistent time-integration is performed using the Euler backward scheme. The predictions of the FE models are compared and the study shows good performance of the mixed formulation. INTRODUCTION Recent years have seen the development of several formulations for composite beams with partial interaction including displacement-based, force-based and mixed formulations [Daniel and Crisinel 1993], [Aribert, Ragneau et al. 1993], [Salari 1999], [Ayoub 2003]. Displacementbased F.E. beam models are derived from the principle of virtual work and provide an approximate solution where the displacement field fulfills strictly the compatibility relations but the equilibrium is satisfied only in a weak sense. For highly nonlinear problems, this formulation

requires a large number of elements [Spacone, Filippou et al. 1996]. The force-based formulation for the composite beam has become particularly attractive in recent years as this formulation seems to be more accurate than the displacement-based formulation with the same number of elements. In this formulation, the strong form of equilibrium equations is satisfied along each element regardless of the section behaviour while compatibility is is enforced in a weak form. The little disadvantage of the force-based elements is the relative complexity to implement such elements in a general purpose finite element program [Spacone, Ciampi et al. 1996]. A two-field mixed formulation is derived from the Hellinger-Reissner variational principle where both displacements and internal forces are varied separately. In the mixed formulation presented by Ayoub [Ayoub 2001], continuity of the displacement field is enforced. In this paper, we analyze the performance of these three finite element formulations for continuous composite beams analysis. For all these formulation, the possibility to have discrete shear connection is considered in this paper. Further, the nonlinear behaviour of steel and concrete is modeled using the rigorous framework of the theory of plasticity. The nonlinear discrete governing equations are derived by applying the standard backward Euler scheme. The study highlights the superiority of the force-based and mixed formulations over the displacement formulation. GOVERNING EQUATIONS OF COMPOSITE BEAMS In this section we recall the field equations for a composite beam with partial shear interaction in a small displacement setting. The constitutive relations are presented in the next section. All variables subscripted with c belong to the concrete slab section and those with s belong to the steel beam. Quantities with subscript sc are associated with the shear connectors.

pe

M c + dM c N c + dN c

Mc

Nc

Hc

y

Tc

x

z

Dsc

Tc + dTc

M s + dM s

Ms

N s + dN s

Ns Ts

dx

Mcst N st c Tcst Qst

Hs

M sst

N sst

Tsst

Ts + dTs

Fig. 1 – Free body diagram of an infinitesimal composite beam segment. Equilibrium The equilibrium conditions are derived by considering the infinitesimal beam segment without connector and the connector elements shown in Figure 1. The equilibrium equations can be written as follows:

where D( x ) = [Ns

∂D − ∂ sc Dsc − Pe = 0

(1)

Q st − bTscQst = 0

(2)

M ] ; M = Mc + Ms ; H = Hc + Hs ; Pe = {0 0 pe } T

Nc

T

and the operator ∂ ,

∂ sc and bsc are defined as: ⎡d ⎢ ⎢ dx ⎢ ∂=⎢ 0 ⎢ ⎢ ⎢⎣ 0

0 d dx 0

⎤ 0 ⎥ ⎥ d ⎤ 1⎤ ⎡ ⎡ ⎥ and bsc = ⎢1 -1 0 ⎥ ; ∂ sc = ⎢1 -1 H ⎥ dx ⎦ H ⎥⎦ ⎣ ⎣ ⎥ d2 ⎥ − 2⎥ dx ⎦

Kinematic relationships The curvature and the axial deformation at any section are related to the beam displacements through kinematics relations. Under small displacements and neglecting the relative transverse displacement between the concrete slab and the steel beam, these relations are as follows:

where d = [us

uc

∂d − e = 0

(3)

∂ sc d − d sc = 0

(4)

v ] is the displacement vector; e = [ε s T

ε c κ ] is the deformation vector, in T

which u is the longitudinal displacement, v the transversal displacement, ε c the strain at the concrete section centroid, ε s the strain at the steel section centroid, κ the curvature and d sc the relative slip between the concrete slab and the steel beam.

Force-deformation relationships The fiber discretization is used to describe the section behaviour in the proposed model. The force-deformation incremental relationship of the cross section is:

ΔD = k Δe or Δe = f ΔD

(5)

where k, f respectively are the tangent stiffness and flexibility matrix of the cross section obtained by integrating over the cross-section the uniaxial discrete constitutive relations for both steel and concrete. Similarly, the connector force is related to the slip at the interface through the following equation: ΔDsc = ksc Δd sc and ΔQst = kst Δd sc

(6)

where ksc ( x ) is the tangent stiffness of the continuous bond (adherence) along the composite beam and kst ( x ) is the tangent stiffness of discrete bond (shear stud).

DISCRETE CONNECTOR ELEMENT To establish the stiffness matrix of such element, one uses the principle of virtual work. Considering the virtual displacement δ qst = [δ us

δ uc δθ ] of the bond element, the internal T

and external virtual work is given by:

δ Wi = Qst δ dsc = Qst [1 −1 −H ] δ qst

(7)

δ We = δ qTsc Q st

(8)

By substituting (6) into (7) and then setting δ Wi = δ We , one obtains:

Kst Δqst = Q st − bst Qsti −1

(9)

where Kst = bTst kst bst is the connector element tangent stiffness matrix.

DISPLACEMENT-BASED FORMULATION In the displacement-based formulation, the displacements serve as primary variables. The displacement field is assumed to be continuous along the beam:

d = aq

(10)

where a is a matrix of 3 × nd displacement shape functions with nd being the total number of displacement degrees of freedom and q the vector of element nodal displacement. The weighted integral form of the equilibrium equation (1) is derived from the principle of virtual work and takes the form:

∫ δ d ( ∂D − ∂ T

sc Dsc

− Pe ) dx = 0

(11)

L

where δ dT ( x ) are the displacement fields fulfilling the kinematics conditions. By integrating by parts (11) and substituting (5) and (6) into (11), one obtains:

Ke Δq = Q + Q 0 + Q R

(12)

where Ke is the element stiffness matrix at the last iteration, defined as:

∫

Ke = ⎡⎣∂T a k ∂ a + ∂T asc ksc ∂ asc ⎤⎦ dx = 0

(13)

L

Q 0 , which corresponds to the vector of nodal forces due to the distributed load p0 , is given by:

∫

Q 0 = aT Pe dx

(14)

L

and Q R is the vector of nodal resisting forces at the last iteration, defined as:

∫

i −1 ⎤ dx = 0 Q R = ⎡⎣∂T a Di −1 + ∂T asc Dsc ⎦ L

(15)

FORCE-BASED FORMULATION In the force-based formulation, the internal forces serve as primary variables. Typically this element has to be developed without rigid-body displacement modes (Figure 2) as the element flexibility matrix need to be eventually inverted to get the element stiffness matrix [Salari 1999]. In short, the internal forces D and the bond force Dsc are expressed in terms of the element nodal forces Q and the bond forces Q sc using the interpolation functions obtained from the equilibrium conditions. The resulting expression is

D = b Q + c Q sc + D0

(16)

Dsc = bsc Q + c sc Q sc

(17)

where b, c, bsc and csc are the forces interpolation functions. p0

Q 1, q 1

Q 1, q 1

Q 2,q 2

Q 2, q 2

Q 4,q 4

(a)

Q 3, q 3

p0

Q 8,q 8

Q1, q1

Q3 , q 3

Q4 , q 4

Q2 , q2

Q 7,q 7

Q5 , q 5

(b)

Fig. 2 Composite beam element: (a) without rigid body modes; (b) Rigid body modes Using the virtual forces δ D and δ Dsc , the compatibility conditions (3) and (4) may be enforced in the integral form as:

∫ δ D ( ∂d − e ) dx + ∫ δ D ( ∂ T

sc

L

sc d −

dsc ) dx = 0

(18)

L

By substituting (5), (6), (16) and (17) into (18) and after eliminating the nodal virtual forces using arbitrariness arguments, one obtains:

⎡Fbb ⎢FT ⎣ bc

Fbc ⎤ ⎡ ΔQ ⎤ ⎡q⎤ ⎡qr ⎤ = − Fcc ⎥⎦ ⎢⎣ ΔQ sc ⎥⎦ ⎢⎣ 0 ⎥⎦ ⎢⎣qrsc ⎥⎦

(19)

where

∫

∫

Fbb = bT f b dx + bTsc fsc bsc dx L

∫

L

∫

Fbc = b f c dx + bTsc fsc csc dx T

L

(20)

L

∫

∫

Fcc = cT f c dx + cTsc fsc csc dx L

∫

L

∫

∫

i −1 qr = bT e i −1dx + bTsc dsc dx + bT f ΔD0dx L

L

L

(21)

∫

∫

∫

i −1 qrsc = cT e i −1dx + cTsc dsc dx + cT f ΔD0 dx L

L

(22)

L

Solving the second equation of (19) for ΔQ sc and substituting the result into the first equation of (19), the governing equation of the element is obtained as:

Fe ΔQ = q − qR

(23)

where Fe is the element flexibility matrix at the last iteration, defined as: T Fe = Fbb (Fcc ) Fbc −1

(24)

qR is element nodal displacements due to the lack of compatibility conditions, defined as: qR = qr + Fbb (Fcc ) qrsc −1

(25)

In order to introduce this formulation in a displacement based procedure, the flexibility matrix Fe must be inverted to obtain the element stiffness matrix. After inversion, the rigid body modes are added using transformation matrices.

HELLINGER-REISSNER MIXED FORMULATION In the Hellinger–Reissner mixed formulation, both the displacement and the internal forces fields along the element are approximated by independent shape functions. More detailed treatment of this mixed formulation can be found in [Zienkiewicz and Taylor 1989]. The equilibrium (1) and compatibility (3) conditions are enforced in an integral form:

∫ δ d ( ∂D − ∂ T

sc Dsc

− Pe ) dx + δ DT ( ∂d − e ) dx = 0

∫

L

(26)

L

By integrating by parts twice the first term of (26) and substituting (5), (6), (10) and (16) into (26), after eliminating the nodal virtual displacements and forces using arbitrariness arguments, one obtains

G ⎤ ⎡ Δq ⎤ ⎡Q + Q 0 ⎤ ⎡Q r ⎤ = − −Fbb ⎥⎦ ⎢⎣ ΔQ ⎥⎦ ⎢⎣ 0 ⎥⎦ ⎢⎣qr ⎥⎦

⎡0 ⎢GT ⎣

(27)

where

G=

∫ (∂

T

)

a b + ∂Tsc asc bsc dx

L

(

∫

)

∫

i −1 Q r = ∂T a Di −1 + ΔD0 dx + ∂Tsc Dsc dx L

qr =

∫ (∂

T

L

)

(28)

L

∫

a e i −1 + ∂Tsc asc dsc dx + b f ΔD0dx L

Solving the second equation for ΔQ and substituting the result into the first equation of (27), the governing equation of the element is obtained as:

K m Δq = Q + Q 0 + Q Rm

(29)

where

( )

K m = G Fbb

−1

GT

( )

Q Rm = Q r − G Fbb

−1

(30)

qr

MATERIAL MODELLING The general constitutive laws used to represent the stress–strain characteristics of the relevant materials are described in this section. For concrete, we adopt the stress-strain curve proposed by the CEB-FIP code model. The classical theory of plasticity is adopted to write the constitutive relations for both concrete and steel. This requires identifying the analytical functions describing hardening or (and) softening in both tension and compression. This procedure is trivial for steel but quite involved for concrete.

One-dimensional hardening/softening plasticity model The total strain splits into an elastic part ε e and a plastic part ε p :

ε = εe +εp

(31)

The stress σ is related to the elastic strain through Hooke’s law:

σ = E (ε − ε p )

(32)

The general expression of the yield function for one-dimensional problems is given by

f (σ , R ) = σ − (σ y + R ) ≤ 0

(33)

where σ y is the initial elastic threshold. To account for the expansion/contraction of the elastic domain, we introduce a stress-like variable R . The evolution of the elastic domain is governed by the following relationship:

R = h ( p)

(34)

If we have hardening, then h ( p ) is a monotonically increasing function. For concrete, this function has two branches. The flow rule is given by

ε p = λ

⎧+1 ∂f = λ sgn(σ ) with λ ≥ 0 and sgn(σ ) = ⎨ ∂σ ⎩ −1

if x > 0 if x < 0

(35)

We can deduce that λ = ε p . The evolution of the strain-like variable p associated to R is given by:

∂f − p = λ = −λ ∂R

(36)

The flow rule must be supplemented by the complementarity relations:

λ ≥ 0 , f (σ , R ) ≤ 0 , λ f (σ , R ) = 0 The consistency condition is given by:

(37)

f(σ , R ) = 0

(38)

The above equations provide a unified format to deal with rate-independent constitutive models. For each component of the composite section, the function R = h ( p ) must be identified in order to match the uniaxial stress-strain relations for each material. The main advantage of the above formulation is consistent time integration. Concrete model

In compression regions, the stress-strain curve suggested by the CEB-FIB model code 1990 [CEB-FIB 1993] includes a monotonically increasing branch up to a peak value, followed by a descending part that gradually flattens to a constant value equal to zero. The initial portion of the ascending branch is linearly elastic, but at about 30% of the ultimate strength, the presence of microcracks leads to nonlinear behavior, with a reduction in tangent modulus. In the subsequent descending branch, the concrete is severely damaged with prominent cracks. In tension, the effect of the rebars is taken into account (tension-stiffening). In this paper, we suggest to transpose the CEB-FIB model into the plasticity framework where a clear distinction between the plastic strain and the total strain is made. In tension, the initial elastic threshold is σ y+ and the contraction of the elastic domain is governed by the following relation

⎛ 1 R ( p ) = f ct ⎜1 − 2 ⎜⎜ 1 + ( p / pu ) ⎝ + c

(

)

⎞ ⎟ ⎟⎟ ⎠

(39)

In compression, the initial elastic threshold is σ y− and the expansion/contraction of the elastic domain is described by the following relation 1 ⎧ 2 2 2 ζ p ζ p ζ + + − ζ4 q − ζ3 ⎪ 1 2 3 ⎪ ⎪ 1 3 ⎧ ⎡ 2 ⎤⎫ Rc− ( p ) = ⎨ 2 2 2 3 1 ⎛ ⎞ ⎛ ⎞ ⎪ i i i ⎥⎪ ⎢ ⎪ β1 p + β 2 + ⎜ ∑ ηi p ⎟ cos ⎨ arccos ⎜ ∑ηi p ⎟ ∑ μi p ⎬ ⎢⎝ i = 0 ⎥⎪ ⎪ ⎝ i =0 ⎠ ⎠ i =0 ⎪3 ⎣⎢ ⎦⎥ ⎭ ⎪⎩ ⎩

(

)

if p ≥ pˆ if p < pˆ

(40)

In the above relation, fct , β i , ζ i , ηi , μi , pu , pˆ are material parameters. The identification process is detailed in [Nguyen 2008]

Steel model and connector stress-slip model

In the present study, the steel is modeled as an elastic-perfectly plastic material incorporating strain hardening. Specifically, the relationship is linearly elastic up to yielding, perfectly plastic between the elastic limit and the commencement of strain hardening, linear hardening occurs up to the ultimate tensile stress and the stress remains constant until the tensile failure strain is reached. The initial elastic threshold is σ y and the following function describes the evolution of the elastic domain size:

⎧0 ⎪⎪ Rs ( p ) = ⎨H ( p − psy ) ⎪ ⎪⎩0

if p < psy if psy ≤ p ≤ psu

(41)

if p > psu

For the connector, an elastic perfectly plastic model is considered. The above relations are discretized using an implicit scheme. The procedure employed to derive the discrete relations is not described in the paper due to lack of space.

NUMERICAL EXAMPLE

In order to study the performance of the displacement-based, force-based and mixed composite beam elements presented above, a continuous composite beam illustrated in Figure 2 is considered. The beam consists of an IPE400 steel section connected to a reinforced concrete slab 800 mm wide and 120 mm thick by means of shear connectors. 152% shear connection and 1% reinforcing bars are used. The material parameters are given in Figure 2.

Concrete fsu fsy

fcm

0.3 fcm tension

Ec ft

Concrete

compression

ε cm

Esh

Es

ε sh

Steel ,Shear connection

fcm = 38MPa

Steel joints fsy = 355MPa Ec = 210000MPa

ft = 3.8MPa

fsu = 454MPa Esh = 2940MPa

Ec = 29000MPa

ε sh = 0.0169

Reinforcement steel Shear connection fsy = fsu = 500MPa fsy = fsu = 3000kN/m Ec = 210000MPa

Ec = 1000MPa

Fig. 2 – Geometrical characteristics of the investigated beam Figure 3 provides the structural global response, where the load-deflection curves obtained by the displacement-based, force-based and mixed formulation using 8 and 24 elements, respectively, are reported. It can be noted that all three formulations give essentially identical results in the elastic range. However, in the plastic range, the three element types show different performances. From Figure 3, it is evident that both force-based and mixed element provides practically identical results. However, the displacement-based formulation with 24 elements seems to be less accurate than the two other formulations with only 8 elements. The solution with 24 force-based elements cannot be further improved by remeshing. To obtain the load-displacement curve with the same accuracy 192 displacement-based are needed. This is essentially because the force-based and mixed element provides a better estimation of the curvature variation in highly nonlinear response than the displacement-based element [Salari, Enrico et al. 1998].

1100 1000 900

Load P [kN]

800 700 600 500 400

8 displacement-based elements 24 displacement-based elements 8 force-based elements 24 force-based elements 8 mixed elements 24 mixed elements

300 200 100 0

0

1

2

3

4 5 Deflection [cm]

6

7

8

9

Fig. 3 Load – deflection Diagrams Figures 4-7 show, along the beam length, the distribution of bending moment, axial force in the slab, slip and curvature, respectively, obtained by the three formulations using 8 elements. As expected, the displacement-based elements give a discontinuity of the bending moment, axial force and the curvature while the interface slip is continuous. It is understandable because in the displacement-based formulation, the equilibrium conditions are satisfied only in weak sense while the compatibility conditions are satisfied in a strict sense. On the contrary, the force-based elements give a continuity of the bending moment, axial force and a discontinuity of the interface slip because only the equilibrium conditions are satisfied in strict sense. In the mixed formulation, both equilibrium conditions and compatibility conditions of section are satisfied in weak sense, only the compatibility condition at the interface is satisfied in strict sense, this why from the figures 4-7 we can observe that the bending moment and the interface slip are continuous in the mixed elements while the axial force and curvature are discontinuous. It should be noted that because of higher degree of polynomials chosen for the connector force field, the force-based model shows a better accuracy for the internal variables [Alemdar 2001].

-300 200

-200

0

0 Axial Force [kN]

Bending moment [kN.m]

-100

100 200 300

-200 -400 -600

400 -800

500 Dislacement based Force based Mixed

600 700

0

3

6 9 Distance from left support [m]

12

Fig. 4 Bending moment distribution

Dislacement based Force based Mixed

-1000 0

3

6 9 Distance from left support [m]

Fig. 5 Axial force distribution

12

CONCLUSIONS

A class of finite element models for continuous composite beam analysis is presented. This class of elements includes displacement-based, force-based and mixed formulations. The nonlinear behaviour of materials and deformable shear connectors has been formulated using the plasticity framework. In terms of global response, the forced-based and the mixed models predict essentially identical results and both elements show a better precision than the displacement-based elements. In terms of local parameters, the force-based models show greater accuracy for the internal variables (bending moment, axial force, and curvature) but their implementation in the general purpose finite element program is quite complex. The mixed model is less accurate than the forced-based model but it shows comparable results and it is easier to implement in the classical finite element program.

0.6

-6 -3

Curvature [rad/mm]

0.4

Slip [mm]

0.2

0

-0.2

Dislacement based Force based Mixed

-0.4

0

3

6 9 Distance from left support [m]

Fig. 6 Slip distribution

0 3 6 9 Dislacement based Force based Mixed

12

12

15

0

3

6 9 Distance from left support [m]

12

Fig. 7 Curvature distribution

REFERENCES

Alemdar, B. N. (2001). Distributed plasticity analysis of steel building structural systems, Georgia Institute of Technology. PhD. Aribert, J. M., Ragneau, E., et al. (1993). "Développement d’un élément fini de poutre mixte acier-béton intégrant les phénomènes de glissement et de semi-continuité avec éventuellement voilement local " Revue Construction Métallique n° 2: 31-49. Ayoub, A. (2001). "A two-field mixed variational principle for partially connected composite beams." Finite Elements in Analysis and Design 37(11): 929-959. Ayoub, A. (2003). "Mixed formulation of nonlinear beam on foundation elements." Computers & Structures 81(7): 411-421. CEB-FIB (1993). CEB-FIP Model Code 1990: Design Code, Thomas Telford. Daniel, B. J. and Crisinel, M. (1993). "Composite slab and strength analysis. Part I: Calculation procedure." Journal of Structural Engineering 119(1): 16-35.

Nguyen, Q. H. (2008). Modélisation numérique du comportement des poutres mixtes acierbéton. Génie Civil. Rennes, France, Institut National des Sciences Appliquées des Rennes. PhD. Salari, M. R. (1999). Modelling of bond-slip in steel-concrete composite beams and reinforcing bars. Departement of Civil, Environmental and Architectural Engineering, University of Colorado. PhD. Salari, M. R., Enrico, S., et al. (1998). "Nonlinear Analysis of Composite Beams with Deformable Shear Connectors." Journal of Structural Engineering 124(10): 1148-1158. Spacone, E., Ciampi, V., et al. (1996). "Mixed formulation of nonlinear beam finite element." Computers and Structures 58: 71-83. Spacone, E., Filippou, F. C., et al. (1996). "Fiber beam-column model for nonlinear analysis of R/C frames. Part I: Formulation." Earthquake Engineering and Structure Dynamics 25(7): 711742. Zienkiewicz, O. C. and Taylor, R. L. (1989). The finite element method. London, McGraw-Hill.